Rotate around one point, with constrain on z-axis for other point, impossible?
There's actually a very quick way to do this. All you need is 3 bones with an IK constraint. Set the root of the first bone at the center of the cylinder in your diagram and the tip on the edge (where the lower stud would be). That way when the bone rotates the tip will follow the track. Now extrude a bone up to the top of the door (top stud). Finally extrude another bone horizontally (left in the case of your diagram) and clear the parent on it.

Then in pose mode, add an IK constraint to the second bone and select Armature in the Target box and the 3rd bone that we cleared the parent on in the Bone box. Now, when you move the 3rd bone (which will be the control bone) the other two bones will follow, the door will open, and the lower stud will always stay the same distance from the center of the track radius.

Optionally you can add a limit location constraint to the control bone (as I did in the gif below) to ensure that the control bone never moves where it shouldn't.



All you then have to do is parent the door to the second bone (the yellow one with the IK constraint) and everything should be good to go.